Science China Information Sciences

, Volume 53, Issue 11, pp 2230–2239 | Cite as

Lattice-valued modal propositional logic and its completeness

Research Papers


Based on the concept of the complete lattice satisfying the first and second infinite distributive laws, the present paper introduces the semantics of the lattice-valued modal propositional logic. It is pointed out that this semantics generalizes the semantics of both classical modal propositional logic and [0, 1]-valued modal propositional logic. The definition of the QMR 0-algebra is proposed, and both the Boole-typed latticevalued modal propositional logic system B and the QMR 0-typed lattice-valued modal propositional logic system QML* are constructed by use of Boole-algebras and QMR 0-algebras, respectively. The main results of the paper are the completeness theorems of both the system B and QML*.


latticed-valued modal propositional logic modal model QMR0-algebra validity completeness 


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© Science China Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Institute of MathematicsShaanxi Normal UniversityXi’anChina

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